Solving Linear Inequalities

Graphing Linear Inequalities

Inequalities

What is an inequality?

Things to Remember:

-The inequalities that contains the equal signs are closed circles when modeling on the number line.

-The inequalities without the equal sign are open circles when modeling on the number line.

Review:

Equations (means-equal)

Remember when we solved equations we isolated the variable on one side by using inverse operations. You will solve inequalities the same way you solve equation. The only difference in them will be the symbol used.

Multiple Choice

How would you graph the following linear inequality?

y ≥ 4

• Dashed line, shade above

• Dashed line, shade below

• Solid line, shade above

• Solid line, shade below

How are equations and inequalities different?

Equations

Have an = sign

Only have one answer when solved

Inequalities

Don't have and = sign instead use inequality symbols

< (less than)

> (greater than)

≤ (less than or equal to)

≥ (greater than or equal to)

Can have multiple answers when solved

Multiple Choice

How would you graph the following linear inequality?

$$y\ ≤\ \frac{4}{3}x\ -\ 4$$  

• Dashed line, shade above

• Dashed line, shade below

• Solid line, shade above

• Solid line, shade below

Example of an inequality and how to interpret your answer.

Graphing Linear Inequalities is the same as graphing lines.

With the addition of shading above or below. We also have check if it is dotted or solid.

Multiple Choice

Solve the following inequality:

x + 2 > 12

• subtract 2 from both sides and your answer is

  x = 10

• subtract 2 from both sides and your answer is

  x > 10

• subtract 2 from both sides and your answer is

  X < 10

Multiple Choice

Solve 5x > 10

• divide by 5 on both sides and your answer is

  x = 2

• divide by 5 on both sides and your answer is

  x < 2

• divide by 5 on both sides and your answer is

  x > 2

Multiple Choice

Solve 3x + 2 > 8

• subtract 2 from both sides and your answer is

  x > 6

• subtract 2 from both sides and your answer is

  x = 6

• subtract 2 from both sides and then divide by 3 on both sides and your answer is

  x > 2

Multiple Choice

Is it dotted or solid?   $$y<\frac{1}{2}x+5$$  

• dotted

• solid

Multiple Choice

Solve the inequality

3x - 7 < 5

• x < 4

• x < -2

• x = 4

Multiple Choice

Is it dotted or solid?   $$y\ge3x-5$$  

• dotted

• solid

One Rule: If you are multiplying or dividing by a negative number you must switch the direction of the inequality symbol. 

• (ex) Solve -4x > -12

• Divide both sides by -4, and because we are dividing by a negative number, we must reverse the inequality sign

  ⇒ -4x/-4 < -12/-4

  ⇒ x < 3

Multiple Choice

When graphing an inequality, you use a dotted line when you use which symbol?

• <, >

• ≤, ≥

Multiple Choice

Try some:

-2x ≤ 4

• divide by -2 o both sided and your answer will be

  x = -2

• divide by -2 on both sides and you will switch the direction of the inequality since you are dividing by a negative and your answer will be

  x ≥ -2

• divide by -2 on both sides and your answer will be

  x ≤ -2

Multiple Choice

When you graph an inequality, you use a solid line when you use which symbols?

• ≤, ≥

• <, >

Multiple Choice

- 3x < -12

• divide both sides by -3 and your answer is

  x < 4

• x = 4

• divide both sides by -3 and your answer is

  x > 4

  because you divided by a negative and had to flip the inequality symbol

What are the solutions to a linear equality?

• Any point (x,y) that satisfies the inequality (makes it true)

• Any point (x,y) that is in the SHADED region when the inequality is graphed.

• Any point (x,y) that is ON THE LINE when the line is a SOLID LINE.

• Any points ON a dotted line are NOT solutions.

Multiple Choice

Example

x ≥ 3

−2

• multiply both sides by -2 and your answer will be

  x ≤ -6

  because you multiplied by a negative you must switch the inequality

• multiply both sides by -2 and your answer will be

  x ≥ -6

• divide both sides by -2 and you get

  x ≥ -2/3

Multiple Select

Which point(s) are solutions to the inequality shown on the graph? Select all that apply.

• (0,5)

• (-2,2)

• (-3,4)

• (-1,1)

• (-4,0)

Multiple Select

Which point(s) are solutions to the inequality shown on the graph? Select all that apply.

• (-4,3)

• (1,3)

• (4,-5)

• (1,-5)

• (-3,-5)

Multiple Choice

Are the points on the boundary line part of the solution set or not?

• Yes, they are part of the solution set

• No, they are NOT part of the solution set

Multiple Choice

Which of the following inequalities matches the given graph?

• y ≥ 2/5x + 1

• y ≥ -2/5x + 1

• y≤  2/5x + 1

• y ≥ -2/5x - 1

Multiple Choice

Which point below is NOT part of the solution set?

• (0, 0)

• (-10, 20)

• (3, 5)

• (0, -2)

Multiple Choice

How can I tell if an inequality will have a solid line?

• It has an equal sign.

• The symbol is greater than.

• The symbol has "or equal to."

• You just guess.

Multiple Choice

Which system of inequalities matches the graph?

• y > 2x - 4
  y < 2
• y < 2x - 4
  y < 2
• y > 2x - 4
  y <  2 
• y < 2x - 4
  y <   -2

Multiple Choice

Which system of linear inequalities is represented by the graph?

• y≤2x-1
  y>-2x+3
• y>2x-1
  y≤-2x+3
• y<2x-1
  y≥-2x+3
• y≤2x-1
  y>2x+3

Multiple Choice

Is (-2,4) a solution to the system?

• Solution
• Not a solution

Multiple Choice

Is (0,0) a solution to the system?

• Solution
• Not a solution

Multiple Choice

Which point is a solution?

• (0, 6)
• (-3, 4)
• (1, 0)
• (-4, 3)

Multiple Choice

Which of the following is not a solution to this system of inequalities?

• (0,-1)
• (0,3)
• (4,0)
• (6,-2)

Multiple Choice

Which of the following is a solution to the systems of inequalities?

• (0, -2)

• (-3, 0)

• (-3, 2)

• (1, 1)